The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

Spectrum of states in icosahedral structures with gN electronic configuration (n = 1– 7). 2. Ab initio calculation of the c20 (I h) molecule and its anions

Ab initio 4- 31G calculations of the C₂₀ (I_h) molecule and its onions in different spectroscopic states arising from the ground electronic configuration g^N are reported. The purpose of these calculations is an independent (quantum chemical) verification of the invariant expansions for the energy obtained in the previous work [15], in particular, of the conclusion derived from these expansions that states with different symmetries in the g^N (I, I_h) configuration are degenerate. Translated fromZhumal Struktumoi Khimii, Vol. 38, No. 4, pp. 605–615, July–August, 1997.     

Some remarks on the semiempirical one-center electron repulsion integrals

The identification of the stage of ionization for various kinds of one-center electron repulsion integrals, occurring when nonbonding or lone-pair electrons are considered explicitly as well as π-electrons, is discussed for conjugated organic molecules containing heteroatoms N. It is concluded that the value for the negative ions should be used for (π_Cπ_C | π_Cπ_C) in all the states but for (π_Nπ_N | π_Nπ_N) only in the π-π states. In the n-π states, the appropriate value of (π_Nπ_N | π_Nπ_N) is that of the neutral atom if the molecule contains only one N atom. If more than one N atom is involved in the molecule, some weighted mean of the values for the negative ion and for the neutral atom should be used. The value for the neutral atom is most adequate for one-center repulsion integrals other than the (ππ | ππ) type in both the π-π and the n-π states. The method of determining these integrals is also discussed. It is concluded that they are to be determined from the consideration of appropriate electron-transfer reactions except for exchange integrals. The exchange integrals are shown to have to be determined from the Slater–Condon parameters derived from the analysis of the experimental atomic energy levels. Illustrative calculations are given for the lower singlet levels of the formaldehyde, pyrazine, pyridine, and the p-benzoquinone molecule. It is found that the calculated energies of the n-π transitions become much too low unless the (ππ | ππ) values of the heteroatoms in the molecule are chosen differently in the n-π states and in the π-π states as pointed out theoretically in this article.     

Extension of SCF and DFT versions of chemical Hamiltonian approach to N interacting subsystems and an algorithm for their efficient implementation

We extended Mayer's chemical Hamiltonian approach methods to N interacting subsystems at the self-consistent field and density functional levels of theory and discussed an efficient algorithm for the implementation. As an example, the energetics of linear (HF)_n chains (n varies from 2 to 8) were calculated. © 1997 by John Wiley & Sons, Inc.     

The extended Koopmans' theorem Fock operator

By expanding the wave function of a system of N particles in terms of products of functions of one and (N-1) particles, the one-particle, nonlocal operator F?_EKT (extended Koopmans' theorem) is determined. It is shown that although this operator is nonhermitian, its eigenvalues and eigenfunctions represent the ionization energies and occupied orbitals, respectively. The eigenfunctions of F?_EKT are the one-particle functions that enter into the expansion of the wave function of the system as partners of the (N-1)-particle wave functions. The eingenvalues are also one-particle energies that, multipled by the orbital occupancy probalities, enter the expression for the total N-particle energy of the system.     

Theoretical Design of High‐spin Organic Molecules with —·N—N— as a Spin‐containing Fragment and Heterocycles as an End Group

Novel stable high spin molecules possessing three different arranged fashions are designed with –·N–N< as a spin‐containing (SC) fragment, various aromatic, such as benzene ( 1 ), pyridine ( 2 ), pyridazine ( 3 ), pyrimidine ( 4 ), pyrazine ( 5 ), triazine ( 6 ) as end groups (EG) and phenyl as a ferromagnetic coupling (FC) unit. The effects of a different end groups on the spin multiplicities of the ground states and their stabilities were investigated by means of AM1‐CI approach. It has been found that the spin densities on the two atoms of the SC fragment are different from delocalization resulting in the specific stability of –·N–N<. In these molecules, the stabilities of the triplet states decrease when the distance between the atoms of central SC (–N–) increases. The orders of the stability of triplet states for 1an , 1bn , 1cn [They are isomers in which SC is connected with FC in different way ( 1an , N₁NNN₁; 1bn , N₁N N₁N; 1cn , NN₁N₁N) and six heterocycles are EG] show that the stability of triplet states with heterocycles as end groups is higher than that with phenyl as end groups, and in the order:triazine (EG)>pyrimidine, pyrazine>pyridine, pyridazine.     

Broken symmetry in valence molecular region within Hartree-Fock calculations

Symmetry restricted and unrestricted Hartree-Fock calculations at theab initio LCAO-MO-SCF level have been carried out on the ground, core and valence hole states of N₂ at various N-N distances. A one-particle criterion for symmetry breaking is discussed. Strong broken-symmetry effects in the inner valence molecular region of N₂ have been found at larger N-N distances. This breaking of symmetry accompanying the symmetry unrestricted Hartree-Fock calculations of the inner valence hole states at large internuclear separations can be considered to be a common phenomenon with all highly symmetric molecules. The outer valence broken-symmetry effects with N₂ have showed some deviations as compared with these effects in the inner valence and core molecular regions.     

Supersymmetry approaches to the radial bound states of the hydrogen‐like atoms

Various structural possibilities for Al_nN and Al_nN₂ (n = 1–7) neutral and cationic isomers are investigated, using the density functional method of Becke's three‐parameter hybrid exchange functional with the Lee–Yang–Parr nonlocal correlation. Structural optimization and frequency analyses are performed with the basis of 6‐311+G(d) for both the neutrals and cations. The calculations predicted the existence of a number of previously unknown isomers (i.e., Al₅N₂ and Al₆N₂). The resulting geometries show that the nitrogen atom prefers to be trapped and not to be on the periphery. Frequency analyses indicate that the 3‐D Al₅N, which was previous proposed as the ground‐state structure, is in fact a first‐order stationary point with an imaginary frequency at 45i (a₂). The optimized ground‐state structure of Al₅N obtained in this work is a planar configuration with the symmetry of C_2v. The calculated adiabatic ionization potentials in their ground states showed that AlN, Al₂N, Al₃N, and Al₄N₂ clusters are more stable than any others in Al_nN and Al_nN₂ (n = 1–7) species, being consistent with the observed time‐of‐flight (TOF) signal intensities. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Size consistency of an algebraic propagator approach

The size consistency property of a general algebraic propagator method referred to as intermediate-state representation (ISR ) is discussed. In this method intermediate states |Ψ_j constructed by a specific orthonormalization procedure from the set of “correlated excited states” Ĉ_j|Ψ_n^N are used to represent the Hamiltonian Ĥ. Here Ĉ_j denotes a physical excitation operator and |Ψ_n^N is the N-electron ground state. The ISR secular equations are shown to be separable, that is, they decouple into independent (local) sets of equations for a system consisting of noninteracting (separate) fragments. This result follows from a general factorization theorem for the intermediate states. Separability is a sufficient condition for size consistency. © 1996 John Wiley & Sons, Inc.     

Molecular orbital theory of the hydrogen bond. 25. Water–uracil complexes in excited n → π states

Hydrogen bonding of uracil with water in excited n → π* states has been investigated by means of ab initio SCF -CI calculations on uracil and water–uracil complexes. Two low-energy excited states arise from n → π* transitions in uracil. The first is due to excitation of the C₄? O group, while the second is associated with excitation of the C₂? O group. In the first n → π* state, hydrogen bonds at O₄ are broken, so that the open water–uracil dimer at O₄ dissociates. The “wobble” dimer, in which a water molecule is essentially free to move between its position in an open structure at N₃? H and a cyclic structure at N₃? H and O₄ in the ground state, collapses to a different “wobble” dimer at N₃? H and O₂ in the excited state. The third dimer, a “wobble” dimer at N₁? H and O₂, remains intact, but is destabilized relative to the ground state. Although hydrogen bonds at O₂ are broken in the second n → π* state, the three water–uracil dimers remain bound. The “wobble” dimer at N₁? H and O₂ changes to an excited open dimer at N₁? H. The “wobble” dimer at N₃? H and O₄ remains intact, and the open dimer at O₄ is further stabilized upon excitation. Dimer blue shifts of n → π* bands are nearly additive in 2:1 and 3:1 water:uracil structures. The fates of the three 2:1 water:uracil trimers and the 3:1 water:uracil tetramer in the first and second n → π* states are determined by the fates of the corresponding excited dimers in these states.     

N-Representability conditions generated by a one-particle grassmann factor of an antisymmetric function

N-representability conditions for a two-particle density operator implied by positive-semidefiniteness of the projection operator P^N+1(?¹ Λ Ψ^N) are derived and discussed. The operator P^N+1(?¹ Λ Ψ^N) projects onto an (N + 1)-particle antisymmetric function ?¹ Λ Ψ^N, the Grassmann product of a one-particle factor ?¹ and an N-particle factor Ψ^N. The polar subcone ??²_N(g, q) to the set of N-representable two-particle density operators ??²_N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one-particle Hilbert space. The facial structure of the convex set ??²_N exposed by elements of ??²_N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N-representability is found. It consists of the decomposition of the N-particle antisymmetric identity operator onto the mutually orthogonal projection operators.     

The density of interface states and their relaxation times in Au/Bi4Ti3O12/SiO2/n‐Si(MFIS) structures

Bismuth titanate (Bi₄Ti₃O₁₂) (BTO) thin films were fabricated on an n‐type Si substrate and annealed by rapid thermal annealing methods. The I‐V measurement shows that the device has properties of Schottky diode with the Φ_B0 of 0.76 eV, n of 2.42, and leakage current of about 10^?7 A at ? 8 V. The experimental C‐V‐f and G/w‐V‐f characteristics of metal‐ferroelectric‐insulator‐semiconductor (MFIS) structures show fairly large frequency dispersion especially at low frequencies due to interface states N_ss. The energy distribution of (N_ss) has been determined by using the high‐low frequency capacitance (C_HF? C_LF) and conductance method. The relaxation time (τ) of interface states was calculated from the conductance method. It has been shown that both the N_ss and relaxation time increase almost exponentially with bias, which activates traps located at deeper gap energies. Copyright © 2011 John Wiley & Sons, Ltd.     

The performance of energy extrapolation procedures in truncated averaged coupled-pair functionals

Summary Energy extrapolation techniques in conjunction with individual configuration selection are applied to averaged coupled-pair functional expansions. In order to test the quality of this approach, benchmark calculations have been performed for N₂, the open and ring forms of O₃, and for the ground and several excited states of CuH and PdH. Reliable energy estimates are obtained for N₂ and the two transition metal hydrides and spectroscopic properties are in close agreement with the values for the non-truncated expansions. In the case of O₃ the perturbation corrections substantially underestimate the complete singles and doubles results. These deviations cancel to a large extent, however, in the calculated isomerization energy. The accuracy of the one-particle density matrix is examined by computing dipole moments for several electronic states of CuH and PdH. Deviations are significant in some cases. For the evaluation of properties the current approach requires modifications.Dedicated to Prof. W. Kutzelnigg on the occasion of his sixtieth birthday     

N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

A theoretical study of the excited‐state decay of acylhydrazones

Acylhydrazones is a novel yet underexploited class of molecular switches. In the present paper, we investigated the excited‐state decay of three model systems of acylhydrazones in the gas phase by a combination of electronic structure calculations and Tully's surface hopping dynamic simulations. Our computational results demonstrated that the S₂(n_Nπ*) state decay of the three model systems leads to both the imine‐like photo‐isomerization through the S₁(n_Nπ*)/S₀ intersection and population of the S₁(n_Oπ*) state that will cross to the triplet manifold. The position of phenyl substituent was found to have an effect on the ratio of the two S₁ states. The present theoretical work provides some understandings of the intramolecular mechanism for de‐population of the excited electronic states of acylhydrazones.     

Correlation entropy of the H2 molecule

Correlation of a quantum many-body state makes the one-particle density matrix nonidempotent. Therefore, the Shannon entropy of the natural occupation numbers measures the correlation strength on the one-particle level. Here, it is shown how this general idea of a correlation entropy must be adapted for two-electron systems in view of conservation laws which mix Slater determinants even in the noninteracting limit. Results are presented for the correlation entropy s of H₂ as a function of the nucleus-nucleus separation R. In the ground state, the entropy of the spatial factor of the wave function maximizes 1.7 bohr beyond the Coulson-Fischer separation. The role of the correlation entropy in density functional theory is also discussed. © 1997 John Wiley & Sons, Inc.     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

Analytic perturbation theory and localization phenomena

This paper attempts to apply the results of analytic perturbation theory to systems exhibiting some degree of randomness in their state variables. In particular, we focus on the localization problem in the Anderson model. It is shown that this approach proves the existence of bound states as a result of the statistical treatment of the Weinberg–van Winter–Hunziker irreducible N-particle kernel. The analysis is extended beyond the one-particle cluster approach and the occurrence of higher-order clusters is discussed in relation to the diluteness criterion and the conditions set on the interaction potential V.     

The origin of energy functional in Roothaan open shell SCF theory

Different methods of averaging of energy over the states of electronic configurations γ^N (n_γ = 1, 2, 3 and N = 1, 2, …, 2n_γ ? 1) leading to Roothaan' energy expression are considered. The consequent values of vector coupling coefficients (VCC ) in energy functionals for various states as well as for average values of energy are presented. It is shown also that in molecular systems of cubic and tetragonal symmetry having electronic configurations t^N (N = 2–4) and e² there exist states for which VCC are dependent on the choice of basis set of degenerate open-shell molecular orbitals. The origin of such “non-Roothaan” terms and peculiarities of its calculation by the restricted Hartree–Fock method are discussed.